Cryptography Reference
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equations:
minfP(A)g = 0
maxfP(A)g = A
P(A) = A=2
minfP(A 1 ;A 2 )g = minfA 1 ;A 2 g
maxfP(A 1 ;A 2 )g = maxfA 1 ;A 2 g
P(A 1 ;A 2 ) = A 1 =2 + A 2 =2
(11.2)
According to Figure 11.3, it is easy to verify the following equations:
A W;a1 = P(A 1 ) + P(A 2 ) + P(A 3 ) + A 4
A B;a1 = P(A 1 ) + P(A 2 ) + P(A 3 ) + A 0 1 + A 0 2 + A 0 3 + A 0 4
A W;a2 = P(A 0 1 ;A 0 2 ) + A 0 4
A B;a2 = A 1 + A 2 + A 3 + A 4 + P(A 0 1 ;A 0 2 ) + A 0 3
A W;b1 = P(A 0 1 + A 0 2 ) + A 0 3
A B;b1 = A 1 + A 2 + A 3 + A 4 + P(A 0 1 + A 0 2 ) + A 0 4
A W;b2 = P(A 1 ) + P(A 2 ) + P(A 3 )
A B;b2 = P(A 1 ) + P(A 2 ) + P(A 3 ) + A 4 + A 0 1 + A 0 2 + A 0 3 + A 0 4
(11.3)
The following Theorem 2 shows that the stacked shares with slight mis-
alignment can still recover the secret image. Note that, by saying the secret
image is recovered by its original color, we mean that a black (resp. white)
pixel in the secret image is represented by a black (resp. white) pixel in the
recovered secret image; and by saying the secret image is recovered by its
complementary color, we mean that a black (resp. white) pixel in the secret
image is represented by a white (resp. black) pixel in the recovered secret im-
age. In order to consist with Definition 1 and Definition 2, we generalize the
denition of contrast and average contrast as follows:
h l
2s 2
= hl
2s 2
and =
(11.4)
where 2s 2 is the area of a pixel (including two subpixels) and the definitions
of h, l, h, and l are the same as that in Denition 1 and Denition 2.
Theorem 2 For a misaligned (2; 2)-VCS, denote (d x ;d y ) as the deviation of
the stacked shares, then the secret image can still be recovered if (d x ;d y ) falls
in the regions of R 1 , R 2 , and R 3 in Figure 11.4, where the properties of the
misaligned scheme are shown in Table 11.3.
Proof: Denote l blackarea (resp. h blackarea ) as the maximum (resp. minimum)
black area of a white (resp. black) pixel, denote l blackarea (resp. h blackarea ) as
the average black area of a white (resp. black) pixel, denote h whitearea (resp.
l whitearea ) as the minimum (resp. maximum) white area of a black (resp.
white) pixel, denote h whitearea (resp. l whitearea ) as the average white area of
a black (resp. white) pixel. According to Figure 11.3, we have:
 
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